Downhole motors are frequently used in drilling operations, particularly in deviated wellbores. A common type of motor used for such operations is a progressing cavity-type motor that works on the Moineau principle. This type of a motor is a positive-displacement type which operates on circulated drilling fluid pumped through cavities of an elastomer internal helix stator, which in turn transfers force into rotational power by turning a steel external helix rotor. The rotor rotates inside the stator in an eccentric manner. The eccentric rotary motion of the rotor is converted into bit rotary motion by connecting the lower end of the rotor to the output shaft leading to the bit through a universal joint coupling. These types of motors can have a single lobe rotor rotating inside a two-lobe stator or can have multi-lobe stator/rotor combinations. The rotor has one less lobe than the stator.
Characteristic curves are provided by manufacturers of downhole motors which are typically used by field personnel to compute the operating speed of the motor and, hence, the rotational speed of the bit. The bit speed is of critical importance during the drilling operation. The bit speed and torque of a positive displacement motor are computed using information on flow rate and pressure drop in the motor, and using such data in performance charts provided by the manufacturer. The result is a method which is not an accurate predictor of the rotor speed. It is thus an objective of the present invention to provide a technique to measure the rotor speed of the positive-displacement motor, particularly when such a motor is part of a bottomhole assembly which incorporates measurement-while-drilling tools. Another objective is to provide efficient and cost-effective drilling by accurate knowledge on a real-time basis of the rotational speed of the bit. To accomplish the object of the present invention, measurements of vibrations generated in the bottomhole assembly during the downhole motor operation are employed.
When drilling with a slightly bent collar, violent lateral vibrations can occur. The bending or sag of the drill collar can occur due to an initial bend or curvature in the collar, sag from gravity and compressional loads, particularly in inclined boreholes or when an unbalanced collar is included in the bottomhole assembly. Centrifugally induced bowing of a collar, in combination with its rotation about the borehole's centerline and about its own center, can cause the collar to whirl in a complicated manner that results in chaotic lateral displacements, impacts and friction at the borehole wall. The magnitude of these centrifugally induced unbalanced forces is proportional to collar mass eccentricity and the square of the rotational rate. Whirling can be destructive when the rotation rate of the assembly equals the natural frequency of the shaft in bending. In a drilling environment, wall contact with a borehole restricts bottomhole assembly deflection. Undesirable consequences, such as surface abrasion caused by forward synchronous whirl and fatigue failures which are caused by backward whirl do, in fact, occur. In backward whirl, the drill collar makes a continuous contact with the borehole wall without slip and the collar center rotates about the borehole center at high rotational rates in a direction opposite to the imposed direction of pipe rotation. Forward synchronous whirl involves the same side of the drillstring making contact with the borehole while rotating.
In a progressive-cavity downhole motor, where the rotor rotates inside a stator in an eccentric manner, the rotor is prone to vibrational characteristics associated with whirling. The rotor center rotates at speed many times greater than the output speed of the shaft leading to the bit and the direction of the eccentric motion of the rotor is opposite to the direction of bit rotation. As a result of eccentric motion, the high rotor whirl speed creates large dynamic unbalanced forces which cause large dynamic loads on the bottomhole assembly. Because the fluid flows between the rotor and stator, there is a coupling between the fluid and the rotor inside the stator, with resulting pressure fluctuations or disturbances which are generated in the fluid at frequencies given by the rotor whirl frequency. Thus, the primary pressure signal in the fluid is modulated because of its coupling or linkage to the whirling action of the rotor. The whirl frequency of eccentric motion of the rotor, .omega..sub.mw, is given by .omega..sub.mw =.omega..sub.m N.sub.r where .omega..sub.m represents the output shaft frequency in radians per second and N.sub.r represents the number of rotor lobes in the motor. Frequencies generated by surface pumping equipment, which circulates mud through the downhole motor which impinges on the rotor lobe, cause pressure fluctuations within the mud at a rate given by N.sub.r .omega..sub.s where .omega..sub.s is the frequency of pump strokes in radians per second. Since the pressure fluctuations in the mud and the drillstring are coupled, the primary rotor signal given by .omega..sub.m N.sub.r is modulated by the hydraulic signal N.sub.r .omega..sub.s. The resulting modulated whirl frequency signals are therefore given by N.sub.r (.omega..sub.m +n.omega..sub.s) and N.sub.r (.omega..sub.m -n.omega..sub.s), where n is an integer.
The derivation for the modulated whirl frequency signals in vibration data is illustrated by letting s.sub.1 and s.sub.2 represent two signals with frequencies N.sub.r f.sub.1 and nN.sub.r f.sub.2. If d.sub.1 and d.sub.2 represent the corresponding DC offsets, then EQU s.sub.1 =A.sub.1 sin(2.pi.N.sub.r f.sub.1)+d.sub.1 EQU s.sub.2 =A.sub.2 sin(2.pi.nN.sub.r f.sub.2)+d.sub.2
The modulated signal is, therefore, given by EQU s.sub.1.s.sub.2 =A.sub.1 sin(2.pi.N.sub.r f.sub.1).A.sub.2 sin(2.pi.nN.sub.r f.sub.2)+d.sub.2 A.sub.1 sin(2.pi.N.sub.r f.sub.1)+d.sub.1 A.sub.2 sin(2.pi.nN.sub.r f.sub.2)+d.sub.1 d.sub.2
Therefore, if f.sub.1 =.omega..sub.m /2.pi., f.sub.2 =.omega..sub.s /2.pi., A.sub.m =A.sub.1, A.sub.s =A.sub.2, d.sub.m =d.sub.1 and d.sub.s =d.sub.2 where .omega..sub.m and A.sub.m represent the circular frequency and amplitude of the mud motor signal and .omega..sub.s and A.sub.s are for pump strokes signal, respectively, then EQU s.sub.1.s.sub.2 =0.5A.sub.m A.sub.s [cos{N.sub.r (.omega..sub.m -n.omega..sub.s)}-cos{N.sub.r (.omega..sub.m +n.omega..sub.s)}]+d.sub.s A.sub.m sin(N.sub.r .omega..sub.m)+d.sub.m A.sub.s sin(N.sub.r .omega..sub.s)+d.sub.m d.sub.s
Additionally, if d.sub.m =0 and d.sub.s &gt;0, i.e., the signal due to pump strokes has a nonzero DC component, then the modulated signal has three components: EQU 0.5A.sub.m A.sub.s cos{N.sub.r (.omega..sub.m -n.omega..sub.s)} EQU 0.5A.sub.m A.sub.s cos{N.sub.r (.omega..sub.m +n.omega..sub.s)} EQU d.sub.s A.sub.m sin(N.sub.r .omega..sub.m)
Because of coupling between fluid and the rotor inside the stator, the large dynamic loads on the BHA could cause pressure fluctuations in the fluid given by whirl frequency N.sub.r W.sub.r. Therefore, the primary pressure signal n.omega..sub.s gets modulated by the rotor whirl frequency as (n.omega..sub.s +N.sub.r .omega..sub.m) and (n.omega..sub.s -N.sub.r .omega..sub.m). The derivation of this result in the pressure data is obtained by letting s.sub.1 and s.sub.2 represent two signals with frequencies N.sub.r f.sub.1 and nf.sub.2. If d.sub.1 and d.sub.2 represent the corresponding DC offsets, then EQU s.sub.1 =A.sub.1 sin(2.pi.N.sub.r f.sub.1)+d.sub.1 EQU s.sub.2 =A.sub.2 sin(2.pi.nf.sub.2)+d.sub.2
Therefore, if f.sub.1 =.omega..sub.s 2.pi., f.sub.2 =.omega..sub.m /2.pi., A.sub.s =A.sub.1, A.sub.m =A.sub.2, d.sub.s =d.sub.1 and d.sub.m =d.sub.2 where .omega..sub.s and A.sub.s represent the circular frequency and amplitude of the pump strokes signal and .omega..sub.m and A.sub.m represents the corresponding values for the motor signal. The modulated signal as before is given by: EQU s.sub.1.s.sub.2 =0.5A.sub.s A.sub.m [cos(n.omega..sub.s +N.sub.r .omega..sub.m)-cos(n.omega..sub.s -N.sub.r .omega..sub.m)]+d.sub.m A.sub.s sin(n.omega..sub.s)+d.sub.s A.sub.m sin(N.sub.r .omega..sub.m)+d.sub.s d.sub.m
In this case, the modulated pressure signal has four components: EQU 0.5A.sub.s A.sub.m cos(n.omega..sub.s -N.sub.r .omega..sub.m) EQU 0.5A.sub.s A.sub.m cos(n.omega..sub.s +N.sub.r .omega..sub.m) EQU d.sub.m A.sub.s sin(n.omega..sub.s) EQU d.sub.s A.sub.m sin(N.sub.r .omega..sub.m)
Other additional frequencies can also be detected on the plots of elapsed time vs. frequency which are called spectrograms. Modulated frequencies .omega..sub.m N.sub.r -.omega..sub.c due to rotor whirling, bit rotational frequency .omega..sub.m +.omega..sub.c due to output shaft rotation where w.sub.c is the collar rotational frequency, and modulated rotor frequency .omega..sub.m .+-..omega..sub.s due to pump stroke frequency w.sub.s, and other frequencies such as .omega..sub.m, .omega..sub.s, and .omega..sub.c, which can be seen on bending moment or vibration spectrograms. The frequencies .omega..sub.m N.sub.r and .omega..sub.s can also be seen on the pressure spectrogram.
The object of the method is to use vibration and pressure data to pinpoint the frequency due to rotor whirl or modulated rotor whirl due to fluid interaction and then compute rotor speed from a formula associating whirl frequency to rotor speed.